Glossary

Alpha (Finance) — Definition, Formula, and Jensen's Alpha

Alpha measures a portfolio's excess return relative to a benchmark predicted by CAPM. Definition, Jensen's alpha formula, worked example, and benchmark-choice limitations.

By NakedPnL Research·May 7, 2026·4 min read

TL;DR

Alpha is the portion of a portfolio's return not explained by its exposure to a benchmark.
Jensen's alpha is the intercept from a CAPM regression of excess portfolio returns on excess market returns.
The reported alpha depends entirely on the benchmark and the regression window — neither is canonical.

Definition

In finance, alpha is the portion of a portfolio's return that cannot be explained by its systematic exposure to a chosen benchmark. The most common formal definition is Jensen's alpha, introduced by Michael C. Jensen in 1968: the intercept term in a regression of a portfolio's excess return on the excess return of a market benchmark, under the Capital Asset Pricing Model (CAPM). A positive alpha indicates the portfolio outperformed the benchmark after adjusting for its beta-driven exposure.

Formula

Jensen's alpha:
  alpha = R_p - [ R_f + beta * (R_m - R_f) ]

or equivalently, the regression intercept in:
  (R_p - R_f) = alpha + beta * (R_m - R_f) + epsilon

where:
  R_p = portfolio return
  R_m = benchmark (market) return
  R_f = risk-free rate
  beta = portfolio's slope coefficient against the benchmark

Multi-factor models (Fama-French three-factor, Carhart four-factor) replace the single market factor with several factors and report a different alpha.

Worked example

Over the same year, a portfolio returns 15%, the benchmark returns 10%, the risk-free rate is 3%, and the portfolio's estimated beta against the benchmark is 1.2. The CAPM-predicted return is 3% + 1.2 × (10% − 3%) = 11.4%. Jensen's alpha is 15% − 11.4% = 3.6 percentage points. Re-run the regression against a different benchmark (e.g. an industry index instead of a broad market index) and the same return stream can produce a noticeably different alpha — sometimes positive, sometimes negative — depending on which factor exposures the new benchmark captures.

Why NakedPnL doesn't display this on profile pages

Alpha is benchmark-dependent and model-dependent. The same return series will produce different alpha figures depending on whether the regressor is a broad market index, a sector index, or a multi-factor model, and on whether the regression is run over 60, 120, or 252 days. There is no canonical choice for a registry of traders running heterogeneous strategies across crypto spot, perpetuals, equities, and prediction markets. Publishing one alpha number on a profile would force NakedPnL to pick a benchmark on the trader's behalf, which would be either misleading (a crypto trader regressed against the S&P 500) or arbitrary. NakedPnL therefore restricts profile-level metrics to TWR, total PnL, and trade count, and exposes the underlying daily-return series via the API so that allocators can run their own factor regressions against benchmarks of their choosing.

Beta — slope coefficient in the CAPM regression
Capital Asset Pricing Model (CAPM)
Information ratio — alpha relative to the volatility of tracking error
Fama-French three-factor model

Frequently asked questions

Is alpha the same as outperformance?

Not exactly. Raw outperformance is portfolio return minus benchmark return. Alpha is the residual after subtracting the portion of return that is mechanically explained by the portfolio's beta exposure to the benchmark. A high-beta portfolio in an up market can show large outperformance with zero alpha.

Can alpha be negative?

Yes. A negative alpha means the portfolio underperformed the CAPM-predicted return given its beta exposure. In a market efficient enough that no skill is rewarded, alpha should be statistically indistinguishable from zero on average across managers.

What is the difference between Jensen's alpha and Treynor's alpha?

Jensen's alpha is an absolute return figure (in percentage points). Treynor's measure is a ratio, dividing excess return over the risk-free rate by beta. The two answer slightly different questions and are usually reported together.

References